Optimal. Leaf size=317 \[ \frac {B d n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 (c+d x)^2 (b c-a d)^2}+\frac {b (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^3 (c+d x) (b c-a d)^2}-\frac {d (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^3 (c+d x)^2 (b c-a d)^2}-\frac {2 A b B n (a+b x)}{g^3 (c+d x) (b c-a d)^2}-\frac {2 b B^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g^3 (c+d x) (b c-a d)^2}+\frac {2 b B^2 n^2 (a+b x)}{g^3 (c+d x) (b c-a d)^2}-\frac {B^2 d n^2 (a+b x)^2}{4 g^3 (c+d x)^2 (b c-a d)^2} \]
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Rubi [C] time = 0.92, antiderivative size = 626, normalized size of antiderivative = 1.97, number of steps used = 28, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac {b^2 B^2 n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{d g^3 (b c-a d)^2}+\frac {b^2 B^2 n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d g^3 (b c-a d)^2}+\frac {b^2 B n \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d g^3 (b c-a d)^2}-\frac {b^2 B n \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d g^3 (b c-a d)^2}+\frac {b B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d g^3 (c+d x) (b c-a d)}-\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d g^3 (c+d x)^2}+\frac {B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 d g^3 (c+d x)^2}-\frac {b^2 B^2 n^2 \log ^2(a+b x)}{2 d g^3 (b c-a d)^2}-\frac {b^2 B^2 n^2 \log ^2(c+d x)}{2 d g^3 (b c-a d)^2}-\frac {3 b^2 B^2 n^2 \log (a+b x)}{2 d g^3 (b c-a d)^2}+\frac {3 b^2 B^2 n^2 \log (c+d x)}{2 d g^3 (b c-a d)^2}+\frac {b^2 B^2 n^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d g^3 (b c-a d)^2}+\frac {b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d g^3 (b c-a d)^2}-\frac {3 b B^2 n^2}{2 d g^3 (c+d x) (b c-a d)}-\frac {B^2 n^2}{4 d g^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^3} \, dx &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d g^3 (c+d x)^2}+\frac {(B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g^2 (a+b x) (c+d x)^3} \, dx}{d g}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d g^3 (c+d x)^2}+\frac {(B (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^3} \, dx}{d g^3}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d g^3 (c+d x)^2}+\frac {(B (b c-a d) n) \int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{d g^3}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d g^3 (c+d x)^2}-\frac {(B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{g^3}-\frac {\left (b^2 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{(b c-a d)^2 g^3}+\frac {\left (b^3 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{d (b c-a d)^2 g^3}-\frac {(b B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{(b c-a d) g^3}\\ &=\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d g^3 (c+d x)^2}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^3 (c+d x)}+\frac {b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d g^3 (c+d x)^2}-\frac {b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d)^2 g^3}-\frac {\left (B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{2 d g^3}-\frac {\left (b^2 B^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{d (b c-a d)^2 g^3}+\frac {\left (b^2 B^2 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{d (b c-a d)^2 g^3}-\frac {\left (b B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{d (b c-a d) g^3}\\ &=\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d g^3 (c+d x)^2}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^3 (c+d x)}+\frac {b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d g^3 (c+d x)^2}-\frac {b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d)^2 g^3}-\frac {\left (b B^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{d g^3}-\frac {\left (b^2 B^2 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{d (b c-a d)^2 g^3}+\frac {\left (b^2 B^2 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{d (b c-a d)^2 g^3}-\frac {\left (B^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{2 d g^3}\\ &=\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d g^3 (c+d x)^2}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^3 (c+d x)}+\frac {b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d g^3 (c+d x)^2}-\frac {b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d)^2 g^3}-\frac {\left (b B^2 n^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{d g^3}+\frac {\left (b^2 B^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{(b c-a d)^2 g^3}-\frac {\left (b^2 B^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{(b c-a d)^2 g^3}-\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{d (b c-a d)^2 g^3}+\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{d (b c-a d)^2 g^3}-\frac {\left (B^2 (b c-a d) n^2\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2 d g^3}\\ &=-\frac {B^2 n^2}{4 d g^3 (c+d x)^2}-\frac {3 b B^2 n^2}{2 d (b c-a d) g^3 (c+d x)}-\frac {3 b^2 B^2 n^2 \log (a+b x)}{2 d (b c-a d)^2 g^3}+\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d g^3 (c+d x)^2}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^3 (c+d x)}+\frac {b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d g^3 (c+d x)^2}+\frac {3 b^2 B^2 n^2 \log (c+d x)}{2 d (b c-a d)^2 g^3}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d (b c-a d)^2 g^3}-\frac {b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d)^2 g^3}+\frac {b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d (b c-a d)^2 g^3}-\frac {\left (b^2 B^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{(b c-a d)^2 g^3}-\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{d (b c-a d)^2 g^3}-\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{d (b c-a d)^2 g^3}-\frac {\left (b^3 B^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{d (b c-a d)^2 g^3}\\ &=-\frac {B^2 n^2}{4 d g^3 (c+d x)^2}-\frac {3 b B^2 n^2}{2 d (b c-a d) g^3 (c+d x)}-\frac {3 b^2 B^2 n^2 \log (a+b x)}{2 d (b c-a d)^2 g^3}-\frac {b^2 B^2 n^2 \log ^2(a+b x)}{2 d (b c-a d)^2 g^3}+\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d g^3 (c+d x)^2}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^3 (c+d x)}+\frac {b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d g^3 (c+d x)^2}+\frac {3 b^2 B^2 n^2 \log (c+d x)}{2 d (b c-a d)^2 g^3}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d (b c-a d)^2 g^3}-\frac {b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d)^2 g^3}-\frac {b^2 B^2 n^2 \log ^2(c+d x)}{2 d (b c-a d)^2 g^3}+\frac {b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d (b c-a d)^2 g^3}-\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{d (b c-a d)^2 g^3}-\frac {\left (b^2 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d (b c-a d)^2 g^3}\\ &=-\frac {B^2 n^2}{4 d g^3 (c+d x)^2}-\frac {3 b B^2 n^2}{2 d (b c-a d) g^3 (c+d x)}-\frac {3 b^2 B^2 n^2 \log (a+b x)}{2 d (b c-a d)^2 g^3}-\frac {b^2 B^2 n^2 \log ^2(a+b x)}{2 d (b c-a d)^2 g^3}+\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d g^3 (c+d x)^2}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d) g^3 (c+d x)}+\frac {b^2 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (b c-a d)^2 g^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d g^3 (c+d x)^2}+\frac {3 b^2 B^2 n^2 \log (c+d x)}{2 d (b c-a d)^2 g^3}+\frac {b^2 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d (b c-a d)^2 g^3}-\frac {b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{d (b c-a d)^2 g^3}-\frac {b^2 B^2 n^2 \log ^2(c+d x)}{2 d (b c-a d)^2 g^3}+\frac {b^2 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{d (b c-a d)^2 g^3}+\frac {b^2 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{d (b c-a d)^2 g^3}+\frac {b^2 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{d (b c-a d)^2 g^3}\\ \end {align*}
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Mathematica [C] time = 0.44, size = 464, normalized size = 1.46 \[ \frac {\frac {B n \left (4 b^2 (c+d x)^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-4 b^2 (c+d x)^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+4 b (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-2 b^2 B n (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+2 b^2 B n (c+d x)^2 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-B n \left (2 b^2 (c+d x)^2 \log (a+b x)+2 b (c+d x) (b c-a d)+(b c-a d)^2-2 b^2 (c+d x)^2 \log (c+d x)\right )-4 b B n (c+d x) (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )}{(b c-a d)^2}-2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d g^3 (c+d x)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.08, size = 654, normalized size = 2.06 \[ -\frac {2 \, A^{2} b^{2} c^{2} - 4 \, A^{2} a b c d + 2 \, A^{2} a^{2} d^{2} + {\left (7 \, B^{2} b^{2} c^{2} - 8 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} n^{2} + 2 \, {\left (B^{2} b^{2} c^{2} - 2 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} \log \relax (e)^{2} - 2 \, {\left (B^{2} b^{2} d^{2} n^{2} x^{2} + 2 \, B^{2} b^{2} c d n^{2} x + {\left (2 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 2 \, {\left (3 \, A B b^{2} c^{2} - 4 \, A B a b c d + A B a^{2} d^{2}\right )} n + 2 \, {\left (3 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n^{2} - 2 \, {\left (A B b^{2} c d - A B a b d^{2}\right )} n\right )} x + 2 \, {\left (2 \, A B b^{2} c^{2} - 4 \, A B a b c d + 2 \, A B a^{2} d^{2} - 2 \, {\left (B^{2} b^{2} c d - B^{2} a b d^{2}\right )} n x - {\left (3 \, B^{2} b^{2} c^{2} - 4 \, B^{2} a b c d + B^{2} a^{2} d^{2}\right )} n - 2 \, {\left (B^{2} b^{2} d^{2} n x^{2} + 2 \, B^{2} b^{2} c d n x + {\left (2 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 2 \, {\left ({\left (4 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} n^{2} + {\left (3 \, B^{2} b^{2} d^{2} n^{2} - 2 \, A B b^{2} d^{2} n\right )} x^{2} - 2 \, {\left (2 \, A B a b c d - A B a^{2} d^{2}\right )} n - 2 \, {\left (2 \, A B b^{2} c d n - {\left (2 \, B^{2} b^{2} c d + B^{2} a b d^{2}\right )} n^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{4 \, {\left ({\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} g^{3} x^{2} + 2 \, {\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} g^{3} x + {\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} g^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 9.30, size = 387, normalized size = 1.22 \[ \frac {1}{4} \, {\left (2 \, {\left (\frac {2 \, {\left (b x + a\right )} B^{2} b n^{2}}{{\left (b c g^{3} - a d g^{3}\right )} {\left (d x + c\right )}} - \frac {{\left (b x + a\right )}^{2} B^{2} d n^{2}}{{\left (b c g^{3} - a d g^{3}\right )} {\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, {\left (\frac {{\left (B^{2} d n^{2} - 2 \, A B d n - 2 \, B^{2} d n\right )} {\left (b x + a\right )}^{2}}{{\left (b c g^{3} - a d g^{3}\right )} {\left (d x + c\right )}^{2}} - \frac {4 \, {\left (B^{2} b n^{2} - A B b n - B^{2} b n\right )} {\left (b x + a\right )}}{{\left (b c g^{3} - a d g^{3}\right )} {\left (d x + c\right )}}\right )} \log \left (\frac {b x + a}{d x + c}\right ) - \frac {{\left (B^{2} d n^{2} - 2 \, A B d n - 2 \, B^{2} d n + 2 \, A^{2} d + 4 \, A B d + 2 \, B^{2} d\right )} {\left (b x + a\right )}^{2}}{{\left (b c g^{3} - a d g^{3}\right )} {\left (d x + c\right )}^{2}} + \frac {4 \, {\left (2 \, B^{2} b n^{2} - 2 \, A B b n - 2 \, B^{2} b n + A^{2} b + 2 \, A B b + B^{2} b\right )} {\left (b x + a\right )}}{{\left (b c g^{3} - a d g^{3}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (d g x +c g \right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.23, size = 861, normalized size = 2.72 \[ \frac {1}{2} \, A B n {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b c d^{3} - a d^{4}\right )} g^{3} x^{2} + 2 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} g^{3} x + {\left (b c^{3} d - a c^{2} d^{2}\right )} g^{3}} + \frac {2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g^{3}} - \frac {2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g^{3}}\right )} + \frac {1}{4} \, {\left (2 \, n {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b c d^{3} - a d^{4}\right )} g^{3} x^{2} + 2 \, {\left (b c^{2} d^{2} - a c d^{3}\right )} g^{3} x + {\left (b c^{3} d - a c^{2} d^{2}\right )} g^{3}} + \frac {2 \, b^{2} \log \left (b x + a\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g^{3}} - \frac {2 \, b^{2} \log \left (d x + c\right )}{{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g^{3}}\right )} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) - \frac {{\left (7 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right )^{2} + 6 \, {\left (b^{2} c d - a b d^{2}\right )} x + 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c d x + 3 \, b^{2} c^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n^{2}}{b^{2} c^{4} d g^{3} - 2 \, a b c^{3} d^{2} g^{3} + a^{2} c^{2} d^{3} g^{3} + {\left (b^{2} c^{2} d^{3} g^{3} - 2 \, a b c d^{4} g^{3} + a^{2} d^{5} g^{3}\right )} x^{2} + 2 \, {\left (b^{2} c^{3} d^{2} g^{3} - 2 \, a b c^{2} d^{3} g^{3} + a^{2} c d^{4} g^{3}\right )} x}\right )} B^{2} - \frac {B^{2} \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )^{2}}{2 \, {\left (d^{3} g^{3} x^{2} + 2 \, c d^{2} g^{3} x + c^{2} d g^{3}\right )}} - \frac {A B \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{d^{3} g^{3} x^{2} + 2 \, c d^{2} g^{3} x + c^{2} d g^{3}} - \frac {A^{2}}{2 \, {\left (d^{3} g^{3} x^{2} + 2 \, c d^{2} g^{3} x + c^{2} d g^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.47, size = 505, normalized size = 1.59 \[ -{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {B^2}{2\,d\,\left (c^2\,g^3+2\,c\,d\,g^3\,x+d^2\,g^3\,x^2\right )}-\frac {B^2\,b^2}{2\,d\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {\frac {2\,A^2\,a\,d-2\,A^2\,b\,c+B^2\,a\,d\,n^2-7\,B^2\,b\,c\,n^2-2\,A\,B\,a\,d\,n+6\,A\,B\,b\,c\,n}{2\,\left (a\,d-b\,c\right )}-\frac {b\,x\,\left (3\,B^2\,d\,n^2-2\,A\,B\,d\,n\right )}{a\,d-b\,c}}{2\,c^2\,d\,g^3+4\,c\,d^2\,g^3\,x+2\,d^3\,g^3\,x^2}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {A\,B}{c^2\,d\,g^3+2\,c\,d^2\,g^3\,x+d^3\,g^3\,x^2}+\frac {B^2\,b^2\,\left (\frac {d^2\,g^3\,n\,x\,\left (a\,d-b\,c\right )}{b}-\frac {d\,g^3\,n\,\left (a\,d-b\,c\right )\,\left (a\,d-2\,b\,c\right )}{2\,b^2}+\frac {c\,d\,g^3\,n\,\left (a\,d-b\,c\right )}{2\,b}\right )}{d\,g^3\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (c^2\,d\,g^3+2\,c\,d^2\,g^3\,x+d^3\,g^3\,x^2\right )}\right )-\frac {B\,b^2\,n\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {2\,a^2\,d^3\,g^3-2\,b^2\,c^2\,d\,g^3}{2\,d\,g^3\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (2\,A-3\,B\,n\right )\,1{}\mathrm {i}}{d\,g^3\,{\left (a\,d-b\,c\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {B^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {2 A B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{g^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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